Question

$\tan \left({\cos }^{-1}\left(\frac{1}{5\sqrt{2}}\right)-{\sin }^{-1}\left(\frac{4}{\sqrt{17}}\right)\right)$ is

• A. $\frac{\sqrt{29}}{3}$
• B. $\frac{29}{3}$
• C. $\frac{\sqrt{3}}{29}$
• D. $\frac{3}{29}$

Answer 1

Answer: (D)

$\tan \left({\cos }^{-1}\left(\frac{1}{5\sqrt{2}}\right)-{\sin }^{-1}\left(\frac{4}{\sqrt{17}}\right)\right)$
$=\tan \left(\frac{\pi }{2}-{\sin }^{-1}\left(\frac{1}{5\sqrt{2}}\right)-{\sin }^{-1}\left(\frac{4}{\sqrt{17}}\right)\right)$
$=\tan \left(\frac{\pi }{2}-\left({\sin }^{-1}\frac{4}{\sqrt{17}}+{\sin }^{-1}\frac{1}{5\sqrt{2}}\right)\right)$
$=\tan \left(\frac{\pi }{2}-{\sin }^{-1}\left(\frac{4}{\sqrt{17}}\sqrt{1-\frac{1}{50}}+\frac{1}{5\sqrt{2}}\sqrt{1-\frac{16}{17}}\right)\right)$
$=\tan \left(\frac{\pi }{2}-{\sin }^{-1}\left(\frac{4}{\sqrt{17}}.\frac{7}{5\sqrt{2}}+\frac{1}{5\sqrt{2}}.\frac{1}{\sqrt{17}}\right)\right)$
$=\cot \left({\sin }^{-1}\frac{29}{5\sqrt{2}.\sqrt{17}}\right)=\cot \left({\cot }^{-1}\frac{3}{29}\right)$
$=\frac{3}{29}$.